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Question 3

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Where is the inverse of a point inside the circle?

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The inverse of a point inside the circle is outside the circle:

the point lies on the ray (AR) going from the center of the circle (A) to the initial point (F)

Justification: Let AR (the radius of the circle of inversion) = 1
:where A is the center of the circle and R is on the circle

Because F is inside the circle, AF is less than 1, but greater than zero

We know: (AF') = (AR)(AR) / (AF)

so (AF') = (1) / (a number less than 1, but greater than zero)

therefore, (AF') is greater than 1, which puts the point F' ( the inverse of F)

outside of the circle

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Where is the inverse of a point outside the circle?

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The inverse of a point outside the circle is inside the circle:

the point lies on the ray going from the center of the circle to the initial point

Justification: Let AR (the radius of the circle of inversion) = 1
:where A is the center of the circle and R is on the circle

Becuase F is outside the circle, AF is greater than 1

We know: (AF') = (AR)(AR) / (AF)

so (AF') = (1) / (a number greater than 1)

therefore, (AF') is less than 1, which puts the point F' ( the inverse of F)

inside of the circle

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Are there points that are their own inverses?

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Yes, points that lie on the circle of inversion are there own inverses

Justification: Let point F be on the circle of inversion
Let AR (the radius of the circle of inversion) = 1

:where A is the center of the circle and R is on the circle

Because F is on the circle AF = AR = 1

We know: (AF') = (AR)(AR) / (AF)

so (AF') = (1) / (1) = 1

therefore, F' ( the inverse of F) must also lie on the circle of inversion

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Where is the inverse of the center of the circle?

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The inverse of the center of the circle is at infinity.

Justification: Let Point F be at point A ( the center of the circle of inversion0
Let AR (the radius of the circle of inversion) = 1

: where A is the center of the circle and R is on the circle

Becuase F is on the center of the circle AF = 0

We know: (AF') = (AR)(AR) / (AF)

so (AF') = (1) / (0) = infinity

therefore, F' ( the inverse of F) is at infinity